9 Arithmetic and Geometric Sequence
|
|
- Herbert Powers
- 7 years ago
- Views:
Transcription
1 AAU - Busness Mathematcs I Lecture #5, Aprl 4, Arthmetc and Geometrc Sequence Fnte sequence: 1, 5, 9, 13, 17 Fnte seres: Infnte sequence: 1,, 4, 8, 16,... Infnte seres: When Gauss was a boy, the teacher ran out of stuff to teach and asked them, n the remanng tme, to compute the sum of all the numbers from 1 to 40. Gauss thought that 1+40 s 41. And +39 s also 41. And ths s true for all the smlar pars, of whch there are 0. So... the answer s 80. One can wonder what would have happened had the teacher asked for the sum of the numbers from 1 to 39. Perhaps Gauss would have noted that 1+39 s 0, as s +38. Ths s true for all the pars, of whch there are 19, and the number 0 s left on ts own. Nneteen 40 s s 760 and the remanng 0 makes 780. Example: Let s consder the seres If we add the frst term to the last we get 0. If we add the second term to the second-to-last we get 0 agan. Now we see that the seres adds up to four 0s, or 80. Now the queston s - wll ths trck work for all seres? If so, why? If not, whch seres wll t work for? Answer: It wll work for all arthmetc seres. The reason that the second par added up the same as the frst par was that we went up by two on the left, and down by two on the rght. As long as you go up by the same as you go down, the sum wll stay the same and ths s just what happens for arthmetc seres. Arthmetc sequence: s a sequence a 1, a,... a n such that a n a n 1 = d for all n. So the dstance between the two followng elements of the sequence s constant. For example: 1,,3,... (d = 1);,4,6, (d = ); 0,3,6, (d = 3) Arthmetc seres: s a sum of elements of arthmetc sequence. The sum s gven by: Σ n =1a = S n = (a 1 + a n ) n Example: Fnd the sum of the followng arthmetc seres: a 1 =1, a 5 =17, d=4, n=5. Σ n =1a = Σ 5 =1a = (a 1 + a n ) n = (1 + 17) 5 = 45 45
2 Example: Now consder the followng sum: Clearly the arthmetc seres trck wll not work here: s not We need a whole new trck. Here t comes. S = where S s the sum we are lookng for. Now, we multply the whole equaton by 3: 3S = Now let s subtract the frst equaton from the second one: S = 4374 whch means that S = 186. Ths trck wll work for all geometrc seres. Geometrc sequence: s a sequence a 1, a,... a n such that between the two followng elements s constant. For example:,4,8,... (r = ); 1,3,9,7,51 (r = 3) a n a n 1 = r for all n. So the rato Geometrc seres: s a sum of elements of geometrc sequence. The sum s gven by: S n = a 1 1 r n 1 r Example: Fnd the followng sum: In ths example, a 1 = 1, a 5 = 16, n = 5, r =. S n = a 1 1 r n 1 r = = 31 Problem: Fnd the sum of the numbers: 3, 7, 11, 15,..., 99. Soluton: In ths example, a 1 = 3, a n = 99, and d = 4. To fnd n we solve the followng equaton: 3 + (n 1)4 = 99 to get n = 5. Then the sum of the numbers s: Σ n =1 = (a 1 + a n ) n = (3 + 99) 5 = 175 Example: Infnte geometrc seres. Fnd the sum of the numbers: 1, 1, 1, In ths case r = 1 < 1. If r < 1 then the sum of nfnte geometrc seres exsts and t can be found as: Σ =1 = a r = 1 1 = 1 (If r > 1 then the sum s equal to nfnty.) 46
3 Example: Fnd a decmal form of the number Notce that: = So r n ths case s 1 10 and hence: Σ =1 = a r = = 7 9 Ths means that 7 9 = Smlarly, = 09 = ; = Problem: Fnd the sum of the frst 30 terms of Soluton: We know that n = 30 and a 1 = 5, and we need the 30th term. Use the defnton of an arthmetc sequence. a 30 = = 11. Therefore, S 30 = 30(5 + 11)/ = Fnancal Mathematcs, Smple and Compound Interest The central theme of these notes s emboded n the queston, What s the value today of a sum of money whch wll be pad at a certan tme n the future? Snce the value of a sum of money depends on the pont n tme at whch the funds are avalable, a method of comparng the value of sums of money whch become avalable at dfferent ponts of tme s needed. Ths methodology s provded by the theory of nterest. A typcal part of most nsurance contracts s that the nsured pays the nsurer a fxed premum on a perodc (usually annual or semannual) bass. Money has tme value, that s, $1 n hand today s more valuable than $1 to be receved one year hence. A careful analyss of nsurance problems must take ths effect nto account. The purpose of ths secton s to examne the basc aspects of the theory of nterest. A thorough understandng of the concepts dscussed here s essental. In ths last context the nterest rate s called the nomnal annual rate of nterest. The effectve annual rate of nterest s the amount of money that one unt nvested at the begnnng of the year wll earn durng the year, when the amount earned s pad at the end of the year Smple and Compound Interest Interest Interest s a fee pad on borrowed captal. The fee s compensaton to the lender for foregong other useful nvestments that could have been made wth the loaned money. Instead of the lender usng 47
4 the assets drectly, they are advanced to the borrower. The borrower then enjoys the beneft of the use of the assets ahead of the effort requred to obtan them, whle the lender enjoys the beneft of the fee pad by the borrower for the prvlege. The amount lent, or the value of the assets lent, s called the prncpal. Ths prncpal value s held by the borrower on credt. Interest s therefore the prce of credt, not the prce of money as t s commonly - and mstakenly - beleved to be. The percentage of the prncpal that s pad as a fee (the nterest), over a certan perod of tme, s called the nterest rate. (wkpeda.org) Smple nterest Smple Interest s calculated only on the prncpal, or on that porton of the prncpal whch remans unpad. The amount of smple nterest s calculated accordng to the followng formula: where A = P (1 + n) A s the amount of money to be pad back P s the prncpal s the nterest rate (expressed as decmal number) n the number of tme perods elapsed snce the loan was taken Smple nterest s often used over short tme ntervals, snce the computatons are easer than wth compound nterest. For example, magne Jm borrows $3,000 to buy a car and that the smple nterest s charged at a rate of 5.5% per annum. After fve years, and assumng none of the loan has been pad off, Jm owes: A = 3000( ) = 935 At ths pont, Jm owes a total of $9,35 (prncpal plus nterest). Compound nterest In the short run, compound Interest s very smlar to Smple Interest, however, as tme goes on dfference becomes consderably larger. The conceptual dfference s that the prncpal changes wth every tme perod, as any nterest ncurred over the perod s added to the prncpal. Put another way, the lender s chargng nterest on the nterest. A = P (1 + ) n In ths case Jm would owe prncpal of $30,060. Generally: If a prncpal P s nvested at an annual rate r (expressed n decmal form) compounded m tmes a year, then the amount A n the account at the end of n years s gven by: ( A = P 1 + ) nm m 48
5 10. Savngs, Loans, Project Evaluatons Tme value of money The tme value of money represents the fact that, loosely speakng, t s better to have money today than tomorrow. Investor prefers to receve a payment today rather than an equal amount n the future, all else beng equal. Ths s because the money receved today can be deposted n a bank account and an nterest s receved. Present value of a future sum where: P V = F V (1 + ) n P V s the value at tme 0 F V s the value at tme n s the rate at whch the amount wll be compounded each perod n s the number of perods Present value of an annuty The term annuty s used n fnance theory to refer to any termnatng stream of fxed payments over a specfed perod of tme. Payments are made at the end of each perod. P V (A) = A 1 (1 + ) + A 1 (1 + ) A 1 (1 + ) = A (1+) n n (1 + ) = A 1 1 (1+) n where: P V (A) s the value of the annuty at tme 0 A s the value of the ndvdual payments n each compoundng perod s the nterest rate that would be compounded for each perod of tme n s the number of payment perods Present value of a perpetuty A perpetuty s an annuty n whch the perodc payments begn on a fxed date and contnue ndefntely. It s sometmes referred to as a perpetual annuty (UK government bonds). The value of the perpetuty s fnte because recepts that are antcpated far n the future have extremely low present value (present value of the future cash flows). Unlke a typcal bond, because 49
6 the prncpal s never repad, there s no present value for the prncpal. The prce of a perpetuty s smply the coupon amount over the approprate dscount rate or yeld, that s P V (P ) = A (1 + ) + A (1 + ) + A (1 + ) = A = A Future value of a present sum F V = P V (1 + ) n Future value of an annuty F V (A) = A(1 + ) (n 1) + A(1 + ) (n ) 1 (1 + )n A = A 1 (1 + ) = A(1 + )n 1 Example: One hundred euros to be pad 1 year from now, where the expected rate of return s 5% per year, s worth n today s money: P V = F V (1 + ) n = = So the present value of 100 euro one year from now at 5% s Example: Consder a 10 year mortgage where the prncpal amount P s $00,000 and the annual nterest rate s 6%. What wll be a monthly payment? The number of monthly payments s n = 10 years 1 months = 10 months The monthly nterest rate s = 6% per year 1 monhs per year = 0.5% per month P V (A) = A 1 1 (1+) n A = P V (A) 1 1 (1+) n (1 + ) n = P V (A) (1 + ) n ( )10 A = ( ) 10 1 = $0.41 per month. 50
7 Example: Consder a depost of $ 100 placed at 10% annually. How many years are needed for the value of the depost to double? F V = P V (1 + ) n 00 = 100( ) n 1.1 n = = ln 1.1 n = ln n ln 1.1 = ln n = ln = 7.7 years ln 1.1 Example: Smlarly, the present value formula can be rearranged to determne what rate of return s needed to accumulate a gven amount from an nvestment. For example, $100 s nvested today and $00 return s expected n fve years; what rate of return (nterest rate) does ths represent? F V = P V (1 + ) n 00 = 100(1 + ) 5 (1 + ) 5 = = (1 + ) = 1/5 = 1/5 1 = 0.15 = 15% Example: A manager of a company has to choose one of two possble projects. Project A requres mmedate nvestment $500 and yelds returns $00, $300, and $400 n the followng three years. For project B t s necessary to nvest $400 now and the expected returns n the next three years are $400, $100 and $50. Supposed that an nterest rate s 10%. Whch project should the manager choose? Havng tme value of money n mnd, manager should choose project wth a hgher present value. P V A = (1 + ) = (1 + ) = 3 = = 30 P V B = (1 + ) = (1 + ) = 3 = = 85 Project A has a hgher present value and hence should be chosen. 51
8 Cash flow: Some of the problems consst of analyzng specal cases of the followng stuaton. Cash payments of amounts C 0, C 1,..., C n are to be receved at tmes 0, 1,..., n. The payment amounts may be ether postve or negatve. A postve amount denotes a cash nflow; a negatve amount denotes a cash outflow. There are 3 types of questons about ths general settng. (1) If the cash amounts and nterest rate are gven, what s the value of the cash flow at a gven tme pont? () If the nterest rate and all but one of the cash amounts are gven, what should the remanng amount be n order to make the value of the cash flow equal to a gven value? (3) What nterest rate makes the value of the cash flow equal to a gven value? Problem: Instead of makng payments of 300, 400, and 700 at the end of years 1,, and 3, the borrower prefers to make a sngle payment of At what tme should ths payment be made f the nterest rate s 6% compounded annually? Soluton: Computng all of the present values at tme 0 shows that the requred tme t satsfes the equaton of value: = t 1.06 t 1400 = = = log 1.06 t = log t = log log Problem: An nvestor purchases an nvestment whch wll pay 000 at the end of one year and 5000 at the end of four years. The nvestor pays 1000 now and agrees to pay X at the end of the thrd year. If the nvestor uses an nterest rate of 7% compounded annually, what s X? Soluton: The equaton of value today s: = X = X X = = = 5737 Thus X = Problem: A three year certfcate of depost carres an nterest rate 7% compounded annually. The certfcate has an early wthdrawal penalty whch, at the nvestors dscreton, s ether a 5
9 reducton n the nterest rate to 5% or the loss of 3 months nterest. Whch opton should the nvestor choose f the depost s wthdrawn after 9 months? After 7 months? Soluton: Frst of all notce that t does not matter at all what s the amount nvested n ths case. The answer wll be the same whether we depost $1 or 350$ or any other amount of money. If the depost s not wthdrawn for the whole 3 years than t would brng: P P ( ) 3 If the depost s wthdrawn after 9 months n whch opton do we lose less? reducton n nterest: P P ( ) 9/1 = 1.037P loss of 3 months nterest: P P ( ) 6/1 = P In ths case reducton n the nterest s better because we get hgher amount of money wth ths opton. If the depost s wthdrawn after 7 months n whch opton do we lose less? reducton n nterest: P P ( ) 7/1 = P loss of 3 months nterest: P P ( ) 4/1 = P In ths case we should choose the second opton - lose 3 months nterest because ths way we get hgher amount of money Bond Prcng In fnance, a bond s a debt securty, n whch the ssuer (borrower) owes the holders (lenders) a debt and s oblged to pay nterest (the coupon) and to repay the prncpal at a later date - maturty. Par Value (as stated on the face of the bond, F ) s the amount that the ssung frm s to pay to the bond holder at the maturty date. Coupon Yeld s smply the coupon payment (C) as a percentage of the face value (F ). Coupon yeld = C/F 53
10 Current Yeld s smply the coupon payment (C) as a percentage of the (current) bond prce (P ). Current yeld = C/P. Yeld to Maturty (YTM) s the dscount rate r whch returns the market prce of the bond. YTM s thus the nternal rate of return of an nvestment n the bond made at the observed prce. Snce YTM can be used to prce a bond, bond prces are often quoted n terms of YTM. Whatever r s, f you use t to calculate the present values of all payouts and then add up these present values, the sum wll equal your ntal nvestment. In an equaton, P = C(1 + r) 1 + C(1 + r) C(1 + r) n + F (1 + r) n = C[1 (1 + ) n ] where C = annual coupon payment (n dollars, not a percent) n = number of years to maturty F = par value P = purchase prce + F (1 + ) n Problem: Suppose your bond s sellng for $950, and has a coupon rate of 7%; t matures n 4 years, and the par value s $1000. What s the YTM? Soluton: The coupon payment s $70 (that s 7% of $1000), so the equaton to satsfy s 70(1 + r) (1 + r) + 70(1 + r) (1 + r) (1 + r) 4 = 950 We are not really gong to solve ths, but the result s that r equals 8.53% (If you want, you can plug ths number back nto equaton to make sure t s correct). Problem: A $5,000 bond pays the holder an nterest rate of 10% payable sem-annually. The bond wll be redeemed at par n 10 years. An nvestor wants to purchase the bond on the bond market to yeld a return of 1% payable sem-annually. What would be the purchase prce of the bond? Soluton: Snce the bond pays 10% on $5,000 semannually, the regular nterest payment wll be: C = = 50 From the nformaton gven, the remanng number of nterest perods s 0. The redempton value of the bond n ten years s the par value or the face value of the bond, $5000. Now to compute the purchase prce, we must calculate the present values of the payments and the redempton value. Snce the yeld rate s the rate the nvestor wants to receve, t s the rate we 54
11 must use to fnd the present values n determnng the purchase prce. Substtutng the values nto our formula, we have: P = C[1 (1 + ) n ] = $, $1, = $4, F (1 + ) n = 50[1 ( ) 0 ] ( ) 0 = Problem: What s the prce of the followng quarterly bond? Face value: $1,000 Maturty: 10 years Coupon rate: 10% Dscount rate: 8% Soluton: /4 [ 1 ] 1 + ( /4) = $ ( /4)
Section 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationTime Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money
Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and
More information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More information10.2 Future Value and Present Value of an Ordinary Simple Annuity
348 Chapter 10 Annutes 10.2 Future Value and Present Value of an Ordnary Smple Annuty In compound nterest, 'n' s the number of compoundng perods durng the term. In an ordnary smple annuty, payments are
More information10. (# 45, May 2001). At time t = 0, 1 is deposited into each of Fund X and Fund Y. Fund X accumulates at a force of interest
1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual e ectve
More informationSection 5.3 Annuities, Future Value, and Sinking Funds
Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme
More information0.02t if 0 t 3 δ t = 0.045 if 3 < t
1 Exam FM questons 1. (# 12, May 2001). Bruce and Robbe each open up new bank accounts at tme 0. Bruce deposts 100 nto hs bank account, and Robbe deposts 50 nto hs. Each account earns an annual effectve
More informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the
More informationLevel Annuities with Payments Less Frequent than Each Interest Period
Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach
More information3. Present value of Annuity Problems
Mathematcs of Fnance The formulae 1. A = P(1 +.n) smple nterest 2. A = P(1 + ) n compound nterest formula 3. A = P(1-.n) deprecaton straght lne 4. A = P(1 ) n compound decrease dmshng balance 5. P = -
More informationSection 2.3 Present Value of an Annuity; Amortization
Secton 2.3 Present Value of an Annuty; Amortzaton Prncpal Intal Value PV s the present value or present sum of the payments. PMT s the perodc payments. Gven r = 6% semannually, n order to wthdraw $1,000.00
More information1. Math 210 Finite Mathematics
1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationTime Value of Money Module
Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the
More informationThursday, December 10, 2009 Noon - 1:50 pm Faraday 143
1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationFINANCIAL MATHEMATICS
3 LESSON FINANCIAL MATHEMATICS Annutes What s an annuty? The term annuty s used n fnancal mathematcs to refer to any termnatng sequence of regular fxed payments over a specfed perod of tme. Loans are usually
More informationEXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR
EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly
More informationFINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals
FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant
More informationMathematics of Finance
5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty;Amortzaton Chapter 5 Revew Extended Applcaton:Tme, Money, and Polynomals Buyng a car
More informationA) 3.1 B) 3.3 C) 3.5 D) 3.7 E) 3.9 Solution.
ACTS 408 Instructor: Natala A. Humphreys SOLUTION TO HOMEWOR 4 Secton 7: Annutes whose payments follow a geometrc progresson. Secton 8: Annutes whose payments follow an arthmetc progresson. Problem Suppose
More informationMathematics of Finance
CHAPTER 5 Mathematcs of Fnance 5.1 Smple and Compound Interest 5.2 Future Value of an Annuty 5.3 Present Value of an Annuty; Amortzaton Revew Exercses Extended Applcaton: Tme, Money, and Polynomals Buyng
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More information8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value
8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest $000 at
More informationANALYSIS OF FINANCIAL FLOWS
ANALYSIS OF FINANCIAL FLOWS AND INVESTMENTS II 4 Annutes Only rarely wll one encounter an nvestment or loan where the underlyng fnancal arrangement s as smple as the lump sum, sngle cash flow problems
More informationSection 2.2 Future Value of an Annuity
Secton 2.2 Future Value of an Annuty Annuty s any sequence of equal perodc payments. Depost s equal payment each nterval There are two basc types of annutes. An annuty due requres that the frst payment
More informationMathematics of Finance
Mathematcs of Fnance 5 C H A P T E R CHAPTER OUTLINE 5.1 Smple Interest and Dscount 5.2 Compound Interest 5.3 Annutes, Future Value, and Snkng Funds 5.4 Annutes, Present Value, and Amortzaton CASE STUDY
More informationIn our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is
Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns
More informationAn Overview of Financial Mathematics
An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take
More informationA Master Time Value of Money Formula. Floyd Vest
A Master Tme Value of Money Formula Floyd Vest For Fnancal Functons on a calculator or computer, Master Tme Value of Money (TVM) Formulas are usually used for the Compound Interest Formula and for Annutes.
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationInterest Rate Forwards and Swaps
Interest Rate Forwards and Swaps Forward rate agreement (FRA) mxn FRA = agreement that fxes desgnated nterest rate coverng a perod of (n-m) months, startng n m months: Example: Depostor wants to fx rate
More informationInterest Rate Futures
Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon
More informationFinancial Mathemetics
Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationOn some special nonlevel annuities and yield rates for annuities
On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationProblem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.
Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When
More informationHedging Interest-Rate Risk with Duration
FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationHollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )
February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs
More informationChapter 15 Debt and Taxes
hapter 15 Debt and Taxes 15-1. Pelamed Pharmaceutcals has EBIT of $325 mllon n 2006. In addton, Pelamed has nterest expenses of $125 mllon and a corporate tax rate of 40%. a. What s Pelamed s 2006 net
More informationChapter 15: Debt and Taxes
Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationSmall pots lump sum payment instruction
For customers Small pots lump sum payment nstructon Please read these notes before completng ths nstructon About ths nstructon Use ths nstructon f you re an ndvdual wth Aegon Retrement Choces Self Invested
More information= i δ δ s n and PV = a n = 1 v n = 1 e nδ
Exam 2 s Th March 19 You are allowe 7 sheets of notes an a calculator 41) An mportant fact about smple nterest s that for smple nterest A(t) = K[1+t], the amount of nterest earne each year s constant =
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationDISCLOSURES I. ELECTRONIC FUND TRANSFER DISCLOSURE (REGULATION E)... 2 ELECTRONIC DISCLOSURE AND ELECTRONIC SIGNATURE CONSENT... 7
DISCLOSURES The Dsclosures set forth below may affect the accounts you have selected wth Bank Leum USA. Read these dsclosures carefully as they descrbe your rghts and oblgatons for the accounts and/or
More informationAS 2553a Mathematics of finance
AS 2553a Mathematcs of fnance Formula sheet November 29, 2010 Ths ocument contans some of the most frequently use formulae that are scusse n the course As a general rule, stuents are responsble for all
More information1 Example 1: Axis-aligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationUncrystallised funds pension lump sum payment instruction
For customers Uncrystallsed funds penson lump sum payment nstructon Don t complete ths form f your wrapper s derved from a penson credt receved followng a dvorce where your ex spouse or cvl partner had
More informationIntra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error
Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor
More informationADVA FINAN QUAN ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS VAULT GUIDE TO. Customized for: Jason (jason.barquero@cgu.edu) 2002 Vault Inc.
ADVA FINAN QUAN 00 Vault Inc. VAULT GUIDE TO ADVANCED FINANCE AND QUANTITATIVE INTERVIEWS Copyrght 00 by Vault Inc. All rghts reserved. All nformaton n ths book s subject to change wthout notce. Vault
More informationHow To Get A Tax Refund On A Retirement Account
CED0105200808 Amerprse Fnancal Servces, Inc. 70400 Amerprse Fnancal Center Mnneapols, MN 55474 Incomng Account Transfer/Exchange/ Drect Rollover (Qualfed Plans Only) for Amerprse certfcates, Columba mutual
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationStaff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall
SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent
More informationA Model of Private Equity Fund Compensation
A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs
More informationConstruction Rules for Morningstar Canada Target Dividend Index SM
Constructon Rules for Mornngstar Canada Target Dvdend Index SM Mornngstar Methodology Paper October 2014 Verson 1.2 2014 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationLIFETIME INCOME OPTIONS
LIFETIME INCOME OPTIONS May 2011 by: Marca S. Wagner, Esq. The Wagner Law Group A Professonal Corporaton 99 Summer Street, 13 th Floor Boston, MA 02110 Tel: (617) 357-5200 Fax: (617) 357-5250 www.ersa-lawyers.com
More informationEffective December 2015
Annuty rates for all states EXCEPT: NY Prevous Index Annuty s effectve Wednesday, December 7 Global Multple Index Cap S&P Annual Pt to Pt Cap MLSB Annual Pt to Pt Spread MLSB 2Yr Pt to Pt Spread 3 (Annualzed)
More informationEffective September 2015
Annuty rates for all states EXCEPT: NY Lock Polces Prevous Prevous Sheet Feld Bulletns Index Annuty s effectve Monday, September 28 Global Multple Index Cap S&P Annual Pt to Pt Cap MLSB Annual Pt to Pt
More informationMorningstar After-Tax Return Methodology
Mornngstar After-Tax Return Methodology Mornngstar Research Report March 1, 2013 2013 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property of Mornngstar, Inc. Reproducton or
More informationInterest Rate Fundamentals
Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"
More informationUnderwriting Risk. Glenn Meyers. Insurance Services Office, Inc.
Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether
More informationUncrystallised funds pension lump sum
For customers Uncrystallsed funds penson lump sum Payment nstructon What does ths form do? Ths form nstructs us to pay the full penson fund, under your non-occupatonal penson scheme plan wth us, to you
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationStock Profit Patterns
Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce
More informationTuition Fee Loan application notes
Tuton Fee Loan applcaton notes for new part-tme EU students 2012/13 About these notes These notes should be read along wth your Tuton Fee Loan applcaton form. The notes are splt nto three parts: Part 1
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationChapter 11 Practice Problems Answers
Chapter 11 Practce Problems Answers 1. Would you be more wllng to lend to a frend f she put all of her lfe savngs nto her busness than you would f she had not done so? Why? Ths problem s ntended to make
More informationTrivial lump sum R5.0
Optons form Once you have flled n ths form, please return t wth your orgnal brth certfcate to: Premer PO Box 2067 Croydon CR90 9ND. Fll n ths form usng BLOCK CAPITALS and black nk. Mark all answers wth
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationProject Networks With Mixed-Time Constraints
Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationYIELD CURVE FITTING 2.0 Constructing Bond and Money Market Yield Curves using Cubic B-Spline and Natural Cubic Spline Methodology.
YIELD CURVE FITTING 2.0 Constructng Bond and Money Market Yeld Curves usng Cubc B-Splne and Natural Cubc Splne Methodology Users Manual YIELD CURVE FITTING 2.0 Users Manual Authors: Zhuosh Lu, Moorad Choudhry
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationA Critical Note on MCEV Calculations Used in the Life Insurance Industry
A Crtcal Note on MCEV Calculatons Used n the Lfe Insurance Industry Faban Suarez 1 and Steven Vanduffel 2 Abstract. Snce the begnnng of the development of the socalled embedded value methodology, actuares
More informationADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET
ADVERSE SELECTION IN INSURANCE MARKETS: POLICYHOLDER EVIDENCE FROM THE U.K. ANNUITY MARKET Amy Fnkelsten Harvard Unversty and NBER James Poterba MIT and NBER Revsed May 2003 ABSTRACT In ths paper, we nvestgate
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationThe Cox-Ross-Rubinstein Option Pricing Model
Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage
More informationNordea G10 Alpha Carry Index
Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and
More information+ + + - - This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12
14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationQuestion 2: What is the variance and standard deviation of a dataset?
Queston 2: What s the varance and standard devaton of a dataset? The varance of the data uses all of the data to compute a measure of the spread n the data. The varance may be computed for a sample of
More informationSUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.
SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976-76-10-00
More informationFixed income risk attribution
5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two
More information